Both instructions on this page tell 3/4 of the story:
Yar is correct: you only have to go to the SFTP settings page to enter your SSH key (which I had to format as a putty key with a .ppk extension - FileZilla will format a normal ssh private key for you if you let it).
ddd is correct that you have to put your login into "Normal" mode (but you still also need the key entered on the SFTP settings).
But there is one more thing you have to have in your settings - you must allow FileZilla to save passwords. This is in the Preferences -> Interface: If you check the box that says "Do Not Save Passwords" you will not be permitted to use Normal mode, and then your ssh won't be utilized - you will be prompted for a password.
So, do all three steps:
- Import SSH Key on the SFTP settings page.
- Do not check the box that says "Do Not Save Passwords" (i.e., allow saving passwords)
- Use Normal mode in the General tab of the server profile settings.